# $T_{0}$-space in which a limit point of a subset whose neighborhood does not intersect the subset at infinitely many points?

Let me formally ask the question again.

Question. Is there a $T_{0}$-space that has a subset $A \subseteq X$ that has the folloing property?

There exists $p \in X$, a limit point of $A$ such that there is a neighborhood of $p$ intersects $A$ at finitely many points.

Note that there is no such subset $A$ in $T_{1}$ space.

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This is the set $\{0,1\}$ with the topology $\{\varnothing,\{1\},\{0,1\}\}$; $0$ is a limit point of $\{1\}$.